TITLE: What are the possible sets of degrees of irreducible polynomials over a field?
QUESTION [38 upvotes]: Hopefully this is not too easy an exercise.
Let $F$ be a field.  Let $I \subset \mathbb{N}$ be the set of all positive integers $d$ such that there exists an irreducible polynomial of degree $d$ over $F$.  What kind of $I$ can occur?
Of course $1 \in I$, and of course we can have $I = \mathbb{N}$ or $I = \{ 1, 2 \}$ or $I = \{ 1 \}$.  The Artin-Schreier theorem implies (I think) that if $I$ is finite, then only the latter two cases occur.  So what kind of infinite $I$ can occur?  
Edit:  For example, correct me if I'm wrong, but we can get $I = \{ 1, p, p^2, ... \}$ for any prime $p$.  Start with $\mathbb{F}_l, l \neq p$, which has absolute Galois group $\hat{\mathbb{Z}} = \prod_q \mathbb{Z}\_q$ and take the fixed field $K$ of $\prod_{q \neq p} \mathbb{Z}\_q$.  Then $G_K = \text{Gal}(\overline{\mathbb{F}_l}/K) = \mathbb{Z}_p$, which (again, correct me if I'm wrong) has the property that its only finite quotients are the groups $\mathbb{Z}/p^n\mathbb{Z}$.  Does this work?

REPLY [14 votes]: This is a very interesting question, however it seems like a characterization of all subsets of $\mathbb N$ which can be such a degree sequence is very hard. I will point you to the paper "On the degrees of finite extensions of a field" by B. Gordon and E.G. Straus where they study this very problem. Unfortunately, it doesn't seem like the paper is available online so I will include here some of their results. For a field $k$, denote by $S(k)$ the sequence of degrees of irreducible polynomials in $k[x]$, since the only finite $S(k)$ are $\{1\}$ and $\{1,2\}$ by Artin-Schreier we will consider infinite sequences.

 Let $P$ be any set of prime numbers, denote by $N_P$ the set of integers whose prime divisors are in $P$. There is a field $k$ of any prescribed characteristic with $S(k)=N_P$.
For every field $k$, if $a,b\in S(k)$ and $gcd(a,b)=1$ then $ab\in S(k)$. If $K$ is algebraic over $k$ then every element of $S(K)$ divides some element of $S(k)$.
For every finite set $A\subset \mathbb{N}$ which doesn't contain $1$, there is a finite $A'\subset \mathbb N$ and a field $k$ so that $S(k)=\mathbb N - A'$ and $A\subset A'$.
If $S(k)$ consists only of powers of a prime $p$ then $S(k)=N_p$.
If $k$ is a CE field (all finite extensions are cyclic) then $S(k)$ is either equal to some $N_P$ or to all elements in some $N_P$ which are not divisible by $4$. 

There is also something you can say from the perspective of computability. If $P$ is $\Sigma_1$ then there is a field $k$ so that $S(k)=N_P$ and you can take $k$ to be computable. See "Sets of degrees of computable fields".